We prove the existence of weak solutions 𝐮:QT→ℝn of the equations of unsteady motion of an incompressible fluid with shear-dependent viscosity in a cylinder QT=Ω×(0,T), where Ω⊂ℝn denotes a bounded domain. Under the assumption that the extra stress tensor 𝐒 possesses a q-structure with q>2nn+2, we are able to construct a weak solution 𝐮∈Lq(0,T;W01,q(Ω))∩Cw([0,T];L2(Ω)) with div𝐮=0. Our approach is based on the Lipschitz truncation method, which is new in this context.

[12] M. De Guzmán, “Differentiation of Integrals in Rn”, Springer-Verlag, Berlin, 1975, with appendices by Antonio Córdoba, and Robert Fefferman, and two by Roberto Moriyón, Lecture Notes in Mathematics, Vol. 481.
MR 457661

[25] O. A. Ladyzhenskaya, New equations for the description of motion of viscous incompressible fluids and solvability in the large of boundary value problems for them, Proc. Steklov Inst. Math. 102 (1967), 95–118.